Given an integer solution $s_m$ to the system,

$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$

$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$

and define the function,

$$F(s_m) = x_1+x_2+\dots+x_n$$

For $n\geq3$, using an *elliptic curve*, it can be shown there are an infinite number of primitive solutions. For $n=4$, we have *positive* and *primitive*,

$$\begin{aligned} s_1 = 10,\, 13,\, 14,\, 44;\quad &F(s_1) = 3^4\\ s_2 = 54,\, 109,\, 202,\, 260; \quad &F(s_2) = 5^4\\ s_3 = 51,\, 65,\, 117,\, 159; \quad &F(s_3) = 2^3 7^2\\ s_4 = 99,\, 315,\, 797,\, 837; \quad &F(s_4) = 2^{11}\\ s_5 = 285,\, 371,\, 547,\, 845; \quad &F(s_5) = 2^{11}\\ s_6 = 815,\, 1297,\, 1781,\, 1939; \quad &F(s_6) = 2^3 3^6\\ s_7 = 259,\, 1307,\, 3485,\, 9349;\quad &F(s_7) = 2^6 3^2 5^2\\ \end{aligned}$$

with $s_6$ and $s_7$ found by MSE user oleg567.

** Questions** (for $n=4$):

- Why do these (apparently smallest) solutions have $F(s_m)$ as a
*smooth number*? - Is this really the complete list with all $x_i<10^4$? (These are the smallest if the list in this post is appropriately complete.)

** Comment**: For $n=3,5$, small solutions do not necessarily have $F(s_m)$ that are smooth numbers.

: Using $s_1$, we have,Edit

$$P_k(x)=(\color{blue}{10}x^2 - x + 88)^k + (\color{blue}{13}x^2 + 68x + 28)^k + (\color{blue}{14}x^2 - 68x + 26)^k + (\color{blue}{44}x^2 + x + 20)^k$$

For $k=3$, it is already a cube,

$$P_3(x) = 45^3(x^2+2)^3$$

For $k=2$, it a *quartic* polynomial to be made a square with a rational point, hence an *elliptic curve*,

$$P_2(x) = y^2$$

with initial points $x = 68,\,165/94$, etc. *All* of the $s_m$ can be used in a similar quadratic form identity. Thus, after removing denominators, the system has an infinite number of integer solutions.

However, we are interested in the behavior of the small positive solutions and which seem to be knowable only by semi-brute force searching.

primitivesolutions]-->what do you mean by "primitive" ? $\endgroup$ – Duchamp Gérard H. E. Apr 26 '15 at 4:15